Problem: Solve for $z$, $ \dfrac{z - 7}{z} = \dfrac{3}{z} - \dfrac{1}{3z} $
Answer: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $z$ $z$ and $3z$ The common denominator is $3z$ To get $3z$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{z - 7}{z} \times \dfrac{3}{3} = \dfrac{3z - 21}{3z} $ To get $3z$ in the denominator of the second term, multiply it by $\frac{3}{3}$ $ \dfrac{3}{z} \times \dfrac{3}{3} = \dfrac{9}{3z} $ The denominator of the third term is already $3z$ , so we don't need to change it. This give us: $ \dfrac{3z - 21}{3z} = \dfrac{9}{3z} - \dfrac{1}{3z} $ If we multiply both sides of the equation by $3z$ , we get: $ 3z - 21 = 9 - 1$ $ 3z - 21 = 8$ $ 3z = 29 $ $ z = \dfrac{29}{3}$